What was that song?

I’m trying to remember the name of a song and the songwriter, but I can’t quite get it. Perhaps one of you might remember the story behind it. I’ve googled the situation to no avail.

The song was a popular ballad in perhaps the late 1940s or early 1950s. A woman sent a letter to a very famous songwriter and said something like “I’ll bet you can’t write a song with this title.” He took up the challenge and wrote a song with that title and used the title as the song’s opening line. It was a huge hit, and he gave her co-writing credit, netting her a lot of money in residuals over the years.

Does this songwriting legend ring anyone’s chimes?

I want to be around (to pick up the pieces) johnny mercer.

Yes! That’s it! The lady was Sadie Vimmerstedt from Youngstown Ohio. Johnny Mercer shared 1/3 of the royalties with her.

Thank you!